A Primer of NMR Theory with Calculations in Mathematica E-Book

Alan J. Benesi

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2015

Beschreibung

Presents the theory of NMR enhanced with Mathematica© notebooks

• Provides short, focused chapters with brief explanations of well-defined topics with an emphasis on a mathematical description
• Presents essential results from quantum mechanics concisely and for easy use in predicting and simulating the results of NMR experiments
• Includes Mathematica notebooks that implement the theory in the form of text, graphics, sound, and calculations
• Based on class tested methods developed by the author over his 25 year teaching career. These notebooks show exactly how the theory works and provide useful calculation templates for NMR researchers

Leseprobe

A PRIMER OF NMR THEORY WITH CALCULATIONS IN MATHEMATICA®

ALAN J. BENESI

Former DirectorNMR FacilityThe Pennsylvania State UniversityUniversity Park, PA, USA

Copyright © 2015 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Benesi, Alan J., 1950–    A primer of NMR theory with calculations in Mathematica® / Alan J. Benesi.        pages    cm    Includes bibliographical references and index.

    ISBN 978-1-118-58899-4 (cloth)1. Nuclear magnetic resonance spectroscopy–Data processing.    I. Title.    II. Title: Primer of nuclear magnetic resonance theory with calculations in Mathematica.    QD96.N8B46 2015    538′.362028553–dc23

                        2014048320

Mathematica® is a registered trademark of Wolfram Research, Inc.

PREFACE

There are two ways to live: you can live as if nothing is a miracle; you can live as if everything is a miracle.

—Albert Einstein

The faint radiofrequency signals detected in nuclear magnetic resonance (NMR) spectroscopy provide a window into the structure and dynamics of atoms in solids, liquids, and gases. No other experimental technique comes close to the range of atomic-level information that NMR can provide. To me, NMR is a miracle. In order to understand NMR, one must master both its experimental and its theoretical aspects. It also helps to be knowledgeable in chemistry. Although experimental NMR is becoming easier as commercial spectrometers evolve, the theory of NMR is still “hard” and is the area in which many NMR spectroscopists are weak. Therefore, in this primer, the theory of NMR is presented concisely and is used in calculations to understand, predict, and simulate the results of NMR experiments. The focus is on the beautiful physics of NMR. The basics of experimental NMR are included to provide perspective and a clear connection with theory. This primer is not comprehensive and is limited to material covered in a graduate-level theoretical NMR class I taught at Penn State for 25 years. There is only cursory discussion of some important NMR topics such as cross polarization or unpaired electron spin–nuclear spin interactions. Nevertheless, a person who has “made it” through this book will be well equipped to understand most topics in the NMR literature.

Throughout my quest to master NMR spectroscopy, I have used the programming language Mathematica, or its predecessor SMP. Here, Mathematica notebooks are used to carry out most of the calculations. These notebooks are also intended to provide useful calculation templates for NMR researchers. Although it is not necessary to have Mathematica to gain understanding from this book, I highly recommend it.

I am grateful to the many pioneers, colleagues, professors, friends, and mentors in the NMR community who have personally or in their publications answered my questions along the way, including but not limited to A. Abragam, H.W. Spiess, M. Levitt, M. Mehring, Burkhard Geil, Paul Ellis, Lloyd Jackman, Juliette Lecomte, Chris Falzone, Ad Bax, Karl Mueller, Richard Ernst, Attila Szabo, Dennis Torchia, Bernie Gerstein, Kurt Wuthrich, Mike Geckle, Clemens Anklin, Matt Augustine, David Boehr, Scott Showalter, John Lintner, Kevin Geohring, Ted Claiborne, Tom Gerig, Tom Raidy, and Alex Pines. The NMR community is lucky to include such kind and inspiring human beings.

Alan J. Benesi

CHAPTER 1INTRODUCTION

Nuclear magnetic resonance (NMR) spectroscopy can provide detailed information about nuclei of almost any element. NMR allows one to determine the chemical environment and dynamics of molecules and ions that contain the observed nuclei. With modern NMR spectrometers, one can observe nuclei of several elements at once. Biological NMR, for example, often employs radio frequency pulses on 1H, 13C, and 15N nuclei within a single experiment. Some of the most useful NMR experiments obtain information by using 20 or more radio frequency pulses applied to the different NMR nuclei at specific times. What makes these sophisticated experiments possible is the mathematical perfection of the quantum mechanics that underlies NMR.

Whether one looks at liquids, solids, or gases, the nuclei being observed are selected by their unique resonance (Larmor) frequencies in the radio frequency range of the electromagnetic spectrum. Choosing a nucleus for observation is analogous to choosing a radio station.

NMR requires a magnet, usually with a very homogeneous magnetic field except when pulsed magnetic field gradients are applied. The magnetic field splits the quantized nuclear spin angular momentum states, thereby allowing transitions between them that can be stimulated by radio frequency excitation. Only transitions between adjacent levels are allowed, and since the levels for a given nucleus are equally separated in energy, the transitions all occur at the same resonance (Larmor) frequency.1 The resonance frequency of a given nucleus is proportional to the strength of the magnetic field and is generally in the radio frequency range of 106 to 109 sec−1 on superconducting magnets of 1–25 Tesla magnetic field strength. Several specific advantages of high magnetic fields are that they give stronger NMR signals, better resolution of chemical shifts, and better resolution for solid samples of odd-half-integer quadrupolar nuclei.

Magnetic resonance imaging is a special type of NMR that takes advantage of the linear relationship between the resonance frequency of a nucleus and the magnetic field. In the presence of a magnetic field gradient, the observed resonance frequency varies with position within the sample, allowing for direct correlation between frequency and position that can be used to create an image. Pulsed magnetic field gradients are also used to select desired NMR signals in nonimaging experiments.

The quantum mechanics that is the basis of NMR spectroscopy has been covered beautifully in books by Abragam (1983), Spiess (1978), Mehring (1983), Ernst et al. (1987), Gerstein and Dybowski (1985), Levitt (2008), and Jacobsen (2007). In this book, the goal is to review the theoretical basis of NMR in a concise, cohesive manner and demonstrate the mathematics and physics explicitly with Mathematica notebooks. Readers are urged to go through all the Mathematica notebooks as they are presented and to use the notebooks as templates for homework problems and for real research problems. The notebooks are a “toolbox” for NMR calculations.

The primer is intended for graduate students and researchers who use NMR spectroscopy. The chapters are short but become longer and more involved as the primer progresses. The primer starts with chapters describing the NMR spectrometer and the NMR experiment and proceeds with the classical view of magnetism, the Bloch equation, and the vector model of NMR. Then it goes directly to quantum mechanics by introducing the density operator, whose evolution can be predicted by using either matrix representation of the spin angular momentum operators or commutation relations between them (product operator theory). It then transitions to coherence order pathways, phase cycles, pulsed magnetic field gradients, and the design of NMR pulse sequences. With the help of Mathematica notebooks, it presents the elegant mathematics of solid state NMR, including spherical tensors and Wigner rotations. Then the focus changes to the effects of atomic and molecular motions in solids and liquids on NMR spectra, including mathematical methods needed to understand slow, intermediate, and fast exchange. Finally, it finishes with the amazing and perfect connection between molecular-level reorientational dynamics and NMR relaxation.

NOTE

1

But higher order transitions can be observed in some cases.

CHAPTER 2USING MATHEMATICA; HOMEWORK PHILOSOPHY

In this primer, the version 8.0.4.0 Mathematica programming language was used to carry out calculations presented in Mathematica notebooks (e.g., xyz.nb). All of the notebooks are provided in a DVD included with the book. It is assumed that the reader has Mathematica and can therefore carry out the calculations step by step or carry them out by evaluating the entire notebook. Step-by-step calculations are advantageous because they enable the user to see the mathematics and learn about the Mathematica language, syntax, and programming at the same time.

The user is urged to make extensive use of the Help→Documentation Center→Search routine to learn about Mathematica. Some useful searches are “Mathematica syntax,” “Mathematica syntax characters,” “Immediate and Delayed Definitions,” and “Defining Variables and Functions.” Once one learns the basics of Mathematica, the notebooks used in this book become almost transparent.

Explanation of the Mathematica programming is presented explicitly in the text when the notebooks are first discussed. These are simply called “Explanation of xyz.nb” at the end of the chapter. The first notebooks and their text explanations are encountered in Chapters 5, 6, 7, and 9. The explanations in the early chapters provide more detailed descriptions of the programming than those in the later chapters.

The user is encouraged to make changes in the provided notebooks and see how they affect the results. It is advisable to go through every calculation in the notebooks step by step, not only to see how physics works in detail but also to learn the Mathematica language and syntax. Be forewarned that crashes can occur, so keep in mind that the correct starting notebook(s) can always be reloaded from the DVD or other storage media.

For those who cannot purchase Mathematica, a free download of the Mathematica CDF Player is available online. This form of Mathematica does not allow the user to change input lines and thereby learn step by step, but it does enable the entire notebook to be evaluated. The Mathematica notebooks (xyz.nb) are also provided as (xyz.cdf) on the DVD provided with the primer.

The homework problems are placed at the end of each chapter. Answers are not provided. The Mathematica notebooks, references, and text explanations provide the necessary help.

CHAPTER 3THE NMR SPECTROMETER

A modern NMR spectrometer consists of a superconducting magnet, a probe that holds the NMR sample in the strongest and most homogeneous part of the magnetic field of the magnet, a console containing radio frequency (rf)–generating electronics, amplifiers, and a receiver; a preamplifier that amplifies the very small NMR signals emitted by the sample after rf excitation; and a computer to control the hardware and process the NMR signals to yield spectra. The rf signals to and from the sample are carried in coaxial cables and propagate at about two-thirds the speed of light. A schematic of a modern NMR spectrometer is shown in Figure 3.1.

Figure 3.1 Schematic of a modern NMR spectrometer.

A superconducting magnet consists of a coil of superconducting wire, typically Niobium–Tin or Niobium–Titanium alloy, immersed in liquid Helium. The boiling temperature of liquid Helium at 1 atm pressure is 4.2 K, well below the superconducting critical temperature of the wire, allowing a current to flow without resistance in the coil. The current flow through the coil generates the magnetic field used in NMR. To accomodate NMR samples at room temperature or other temperatures, the liquid helium–immersed superconducting coil is housed in a toroidal dewar, the central “hole” of which is open to the atmosphere at room temperature and holds the shim stack and NMR probe. Typically, the dewar is constructed of stainless steel, with high vacuum between dewar sections containing liquid Nitrogen and liquid Helium and also between the liquid Nitrogen dewar and the outer surface of the magnet. Figure 3.2 shows a schematic of a vertical cross-section of a superconducting magnet.

Figure 3.2 Vertical cross-section of a superconducting magnet.

Activation of a superconducting magnet is carried out by using an external power supply to ramp up the current in the superconducting coil (already immersed in liquid He) until the desired current and corresponding magnetic field are achieved. At this point, the external power supply is disconnected from the superconducting coil, but the current is maintained in the coil because there is no resistance. As long as the coil is intact and immersed in liquid helium, the current and corresponding magnetic field can be maintained indefinitely.

Unfortunately, the world has used up most of the easily accessible Helium, so efforts are underway to reclaim Helium whenever possible and to develop liquid Nitrogen superconductors that can sustain the high current needed for NMR magnets.

The NMR sample fits in the probe and is situated at the strongest and most homogeneous part of the magnetic field where all of the magnetic lines of force are nearly perfectly parallel and of equal magnitude. The homogeneity of the magnetic field across the sample is further improved by using small corrective electromagnets called shims, located in the “shim stack” that surrounds the cavity occupied by the probe. Modest adjustable currents through the shims allow the magnetic field across the sample to be made almost perfectly homogeneous, thereby increasing both resolution and vertical peak intensity in the NMR spectrum. The NMR sample placement relative to the magnetic lines of force is shown in Figure 3.3.

Figure 3.3 NMR sample placement relative to magnetic lines of force, vertical cross-section with expanded view.

CHAPTER 4THE NMR EXPERIMENT

The sample is placed in the probe in the magnet. The probe is tuned to the desired resonance frequency(ies) and then shimmed to obtain a homogeneous magnetic field, that is, B = {0,0,B0},1 as shown in Figure 3.3. The magnetic field removes the degeneracy of the nuclear spin states—the Zeeman effect. The Zeeman Hamiltonian is ĤZ = −γ B0 Îz. If the nuclear spin quantum number is I, the Zeeman Hamiltonian splits the quantized states into 2I+1 evenly spaced energy levels, ranging from m = −I to +I in units of 1, corresponding to the different expectation values for Îz (I = 0 nuclei such as 16O and 12C have only one level and are not NMR observable). Figure 4.1 shows the Zeeman energy levels for an I = 1/2 spin and an I = 1 spin.